Generally, "rich" is defined as having a lot of money and living a luxurious life. However, for me, being rich means feeling financially secure and at peace, being able to live worry-free, debt-free, and having enough income to meet all my needs and pay my bills.
I feel that with all of the above, I am already rich, happy, and free, living a life filled with gratitude without the burdens, fears, worries, and anxieties that come with having a lot of money and a job.
Everyone has different standards, and there's nothing wrong with what you said: someone feels rich if they have everything fulfilled without the need for luxury. However, for some people wealth is about luxury.
Usually, the standard of wealth continues to rise after we achieve something. That's just how humans are, and I don't deny that, because I've been in a position where the desire for more always arises when I get something.
And being rich is simply about satisfying our desires. That's what really happens when we achieve what we desire. I don't think there's anything else, and we can live by any standard, even if we don't want it.